The non-Hermitian skin effect is an anomalous localization phenomenon induced by nonreciprocal dissipation and has attracted considerable attention in recent years both theoretically and experimentally. In this article, we review the multifractal aspects of the non-Hermitian skin effect. In particular, we discuss how the many-body skin effect exhibits multifractality in many-body Hilbert space, unlike the trivial Hilbert-space occupation of the single-particle skin effect on crystalline lattices. We further highlight that the many-body skin effect can coexist with random-matrix spectral statistics, in contrast to the multifractality associated with many-body localization, which typically accompanies the absence of ergodicity. We also introduce a solvable model on a Cayley tree as an effective description of the many-body Hilbert space, in which the multifractal dimensions can be obtained analytically. This review provides a unified perspective on multifractal structures in the non-Hermitian skin effect across single-particle, many-body, and tree models, and clarifies their distinctive relation to ergodicity in open quantum systems.
Comments
19 pages, 6 figures. Submitted as an invited article to Quantum Science and Technology Focus Collection on "Non-Hermitian Quantum Many-Body Physics"
The non-Hermitian skin effect is an anomalous localization phenomenon induced by nonreciprocal dissipation and has attracted considerable attention in recent years both theoretically and experimentally. In this article, we review the multifractal aspects of the non-Hermitian skin effect. In particular, we discuss how the many-body skin effect exhibits multifractality in many-body Hilbert space, unlike the trivial Hilbert-space occupation of the single-particle skin effect on crystalline lattices. We further highlight that the many-body skin effect can coexist with random-matrix spectral statistics, in contrast to the multifractality associated with many-body localization, which typically accompanies the absence of ergodicity. We also introduce a solvable model on a Cayley tree as an effective description of the many-body Hilbert space, in which the multifractal dimensions can be obtained analytically. This review provides a unified perspective on multifractal structures in the non-Hermitian skin effect across single-particle, many-body, and tree models, and clarifies their distinctive relation to ergodicity in open quantum systems.
Non-Hermitian many-body physics reveals a rich interplay between topology, localization, and boundary effects, yet their collective behavior in interacting disordered systems remains largely unexplored. In this work, we study an interacting non-Hermitian spin chain subject to a quasi-periodic longitudinal field, providing a unified and controlled setting, where non-Hermitian dynamics, interactions, and localization mechanisms intertwine. Remarkably, we discover a "D-shaped" many-body mobility edge that separates extended and localized eigenstates, while simultaneously delineating regimes of many-body localization and the many-body skin effect (where many-body eigenstates acquire an anomalous drift towards a boundary under open boundaries) emerging from the combined action of interactions, non-Hermiticity, and driving amplitude. We demonstrate that the skin effect induces multifractal scaling in the non-Hermitian eigenstates, providing a clear signature of the many-body skin effect. Employing diagnostics such as the fractal dimension, complex eigenvalue fractions, and many-body inverse participation ratios, we map out a unified phase diagram in which all measures consistently identify the "D-shaped" mobility edge. Finally, we probe this interplay using both complex level-spacing statistics and dynamical observables such as density imbalance, entanglement growth, and wave-packet evolution, culminating in a rich many-body mobility phase diagram that captures both the many-body skin effect and localization transitions. Our results identify a clear, defining signature of the "D-shaped" many-body mobility edge, and underscore its pivotal role in shaping the physics of open quantum many-body systems.
For decades, Hermiticity was considered an immutable axiom of quantum mechanics, essential for ensuring real energies and unitary evolution. This perspective has shifted radically, driven by the realization that non-Hermitian Hamiltonians provide a powerful effective description of open quantum systems, granting access to unique phenomena such as Exceptional Points and the Non-Hermitian Skin Effect. In this Perspective, we chart the trajectory of this field, moving from its established foundations in single-particle, linear models to the emerging frontier of interacting many-body systems. We first clarify the physical origins of non-Hermitian dynamics, distinguishing between mean-field approximations, conditional "no-click" evolution, and exact Liouvillian dynamics. We then focus on the rich phenomenology arising from the interplay of non-Hermiticity and interactions. We discuss interaction-induced topological phases, the generalization of skin effects to the many-body Hilbert space, and the distinct signatures of dissipative quantum chaos and complexity. Finally, we highlight collective phenomena in nonlinear regimes, including skin solitons and dissipative phase transitions. We also comment on measurement-induced entanglement transitions and their relation to non-Hermitian spectra and topology. By synthesizing these diverse developments, we provide a roadmap for the future of non-Hermitian physics.
The explorations of non-Hermiticity have been devoted to investigate the disorder-induced many-body localization (MBL). However, the sensitivity of the spatial boundary conditions and the interplay of the non-Hermitian skin effect with many-body phenomena are not yet clear. For a MBL system in the presence of nonreciprocal tunnelings and random disorder potential, we identify two different complex-real spectral transitions, one is present for both open and periodic boundaries while the other is present only for open boundaries of coupled non-Hermitian chains. The later is driven due to the interchain coupling at weak disorder where the level statistics of the real eigenenergy phase follows a Gaussian orthogonal ensemble. We further characterize wave functions through the (biorthogonal) inverse participation ratio and fractal dimension, which reveal the suppression of skin effect in the non-Hermitian MBL phase. Finally, we demonstrate that the quench dynamics of the local particle density, spin imbalance, and entanglement entropy also signify the hallmark of the boundary effects and nonergodic character of many-body localization.
Non-Hermiticity greatly expands existing physical laws beyond the Hermitian framework, revealing various novel phenomena with unique properties. Up to now, most exotic nonHermitian effects, such as exceptional points and non-Hermitian skin effects, are discovered in single-particle systems. The interplay between non-Hermitian and manybody correlation is expected to be a more fascinating but much less explored area. Due to the complexity of the problem, current researches in this field mainly stay at the theoretical level. The experimental observation of predicted non-Hermitian manybody phases is still a great challenging. Here, we report the first experimental simulation of strongly correlated non-Hermitian many-body system, and reveal a new type of nonHermitian many-body skin states toward effective boundaries in Hilbert space. Such an interaction-induced non-Hermitian many-body skin effect represents the aggregation of bosonic clusters with non-identical occupations in the periodic lattice. In particular, by mapping eigen-states of three correlated bosons to modes of the designed threedimensional electric circuit, non-Hermitian many-body skin effects in Hilbert space is verified by measuring the spatial impedance response. Our finding not only discloses a new physical effect in the non-Hermitian many-body system, but also suggests a flexible platform to further investigate other non-Hermitian correlated phases in experiments.
The non-Hermitian skin effect is nonreciprocity-induced localization phenomena in which a macroscopic number of eigenstates accumulate anomalously at the boundary, accompanied by the extreme sensitivity to boundary conditions. Here, we develop a geometric characterization of the non-Hermitian skin effect. We demonstrate that the localization length scale associated with the skin effect is encoded in the quantum metric defined solely from right eigenstates, but not in the biorthogonal quantum metric. Moreover, we show that the quantum metrics can exhibit the power-law divergences at gapless points that depend on the different boundary conditions. We also reveal that cusps of the generalized Brillouin zone in non-Bloch band theory are signaled by discontinuities in the quantum metrics. We illustrate these behavior using prototypical non-Hermitian models, such as the Hatano-Nelson model and the non-Hermitian, nonreciprocal Su-Schrieffer-Heeger model.
Non-Hermiticity enables macroscopic accumulation of bulk states, named non-Hermitian skin effects. The non-Hermitian skin effects are well established for single-particle systems, but their proper characterization for general systems is elusive. Here, we propose a general criterion of non-Hermitian skin effects, which works for any finite-dimensional system evolved by a linear operator. The applicable systems include many-body systems and network systems. A system meeting the criterion exhibits enhanced non-normality of the evolution operator, accompanied by exceptional characteristics intrinsic to non-Hermitian systems. Applying the criterion, we discover a new type of non-Hermitian skin effect in many-body systems, which we dub the Fock space skin effect. We also discuss the Fock space skin effect-induced slow dynamics, which gives an experimental signal for the Fock space skin effect.